In spectrometry, an interferogram or free induction decay contains the raw signals from different signal components produced by the excitation method of a spectrometer. Such excitation results in an oscillating signal that is a function of amplitude versus time. Different regions of the interferogram in the time domain contain information that can increase the signal-to-noise ratio or increase the resolution between neighboring peaks of its corresponding spectrum in the frequency domain. The increase in one, however, typically results in a tradeoff of the other.
Since interferograms can only be measured out to some finite time, all experimental interferograms are finite. Upon transforming the finite time-based interferogram into the frequency domain using a technique such as the Fourier Transform, the resulting spectrum includes an observable “ringing,” sometimes referred to as Gibbs phenomenon. This ringing, also described as truncation error, is preferably avoided because it can interfere with the spectral analysis of the signal components. In spectrometry, application of an apodization function to an interferogram signal can help reduce Gibbs phenomenon ringing as well as eliminate unwanted side lobes in the resulting frequency spectra.
FIG. 1 is a diagram illustrating a number of several standard apodization functions and their corresponding waveforms. Standard apodization functions include Bartlett (triangle) 5a, Blackman 5b, Connes 5c, Cosine 5d, Gaussian 5e, Hamming 5f, Hanning 5g, Uniform (boxcar) 5h, Welch 5i, trapezoidal 5j, Norton-Beer (strong, medium and weak), Happ-Genzel, Exponential decay, Gaussian decay, and other single- or double-sided apodization functions known to those skilled in the art.
Apodization functions are used to apply a weight profile to an interferogram prior to its transformation into the frequency domain. Since different sections of the interferogram correspond to signals of different oscillation frequencies, the shape of the corresponding weight profile produced by an apodization function can enhance the signals of certain frequencies and reduce others.
Low frequency signals are typically characterized as having broad spectral peaks that are gaussian in shape. In the case of infrared spectrometry, such broad spectral peaks are several hundreds or thousands of wavenumbers in width. Instrument functions representing the characteristics of a spectrometer and optical response signals resulting from spectrometry of certain solid materials are representative of low frequency signals.
Medium frequency signals are typically characterized as having medium width spectral peaks resembling the lineshape of a Lorentzian-Gaussian mix. In the case of infrared spectrometry, such medium width peaks are several tens of wavenumbers in width. Optical response signals resulting from infrared spectrometry of most solids, liquids and some gases are representative of medium frequency signals. Medium frequency signals can also be associated with instrument functions that represent characteristics of a spectrometer in the form of, for example, artifacts arising from thin film or coating interferences.
High frequency signals are typically characterized as having narrow Lorentzian peaks. In the case of infrared spectrometry, such narrow peaks are several wavenumbers wide or less. These high frequency signals are typically associated with optical response signals resulting from infrared spectrometry of gases. Another contributor to the interferogram that is composed of high frequency components is that of random noise.
Standard apodization functions typically apply a continuously decaying weight profile in which high weights are applied to interferogram points that correspond to the low frequency signals (e.g., low interferogram point numbers) and lower weights are applied, down to zero for some apodization functions, to the latter interferogram points that correspond to high frequency signals. After transformation, the “ringing” around the narrow, high frequency peaks in the resulting spectra can be significantly reduced and the random noise in the system is suppressed.
The choice of the apodization function generally resides in a trade-off between resolution of the signal frequencies and truncation error or artifact ringing. Frequencies that are very close together can be resolved if the interferogram oscillation is allowed to continue until the frequencies begin to separate from each other. In other words, to obtain high resolution, an apodization function that completely suppresses the signals at the high frequency end (i.e., the latter interferogram point numbers) reduces the ability to resolve narrow peaks.
Thus, the apodization function that is typically used for retaining the highest resolution is the boxcar apodization function. This function is the same as multiplying the interferogram by the value of one across most of time decay and then multiplying by zeros at or near the end of the interferogram. The boxcar apodization function provides the best resolution for that particular system but the worst ringing in the frequency domain. Conversely, an apodization function which decays smoothly to zero at the end of an interferogram, such as the cosine apodization function, provides worse resolution but better elimination of the ringing in the frequency domain.
For more information on standard apodization functions, refer to Peter R. Griffiths and James A. de Haseth, Fourier Transform Infrared Spectrometry, Wyley Interscience (1986) (hereinafter “Griffiths et al”), the entire contents of which are incorporated herein by reference.